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The difference of cubes of two consecutive natural numbers is 1657. Find the numbers.

Sagot :

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x, x+1 - two consecutive natural numbers

[tex](x+1)^3-x^3=1657 \\ x^3+3x^2+3x+1-x^3=1657 \\ 3x^2+3x+1=1657 \\ 3x^2+3x-1656=0 \\ 3x^2+72x-69x-1656=0 \\ 3x(x+24)-69(x+24)=0 \\ (3x-69)(x+24)=0 \\ 3x-69=0 \ \lor \ x+24=0 \\ 3x=69 \ \lor \ x=-24 \\ x=23 \ \lor \ x=-24[/tex]

x must be a natural number, so x=23.

[tex]x=23 \\ x+1=23+1=24[/tex]

The numbers are 23 and 24.
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